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During the Olympics the commentators made a big deal about the need to maintain very specific atmospheric conditions in the velodrome, including having airlocks to enter and exit the building.

They stated that the atmosphere is very warm, and humid.

Why are those precise conditions needed for velodrome events?

My suspicion is the high temperature lowers the air density and thus lowers wind resistance but that's purely a guess. I've no idea what the humidity levels would do to help, surely that would increase the air density again!

Also, with such high temperatures, is there a danger of athletes overheating?

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Would be nice to know what very warm and humid means in numbers. My first guess would be, that it is around 25°C and the humidity aids the cooling because sweat can evaporate easier. Another thought is that it is for preventing the material the track is made of from "working" because of the changes in temperature and humidity. If this claim is true at all… –  Baarn Sep 1 '12 at 10:01
    
@WalterMaier-Murdnelch It was a couple of weeks now, so I can't remember exact figures, but I think it was closer to 30 than that (maybe even over). The figure or 60% humidity seems stuck in my mind though. Wouldn't dry air make sweat evaporate more easily? (Apparently it's 28 C) –  GordonM Sep 1 '12 at 11:26
    
@WalterMaier-Murdnelch -- High humidity doesn't make sweat evaporate easier -- quite the contrary, it makes sweat pool on the skin (and drip off) and hampers cooling. Humidity would likely improve tire traction, though. –  Daniel R Hicks Sep 1 '12 at 11:57
    
The theorists tell us that humidity differences, within normal ranges, do not affect air viscosity enough to be significant. However, cyclists who bike in varied conditions will tell you that high humidity seems to make the air more viscous. –  Daniel R Hicks Sep 1 '12 at 11:58
    
Of course, I confused humidity and aridity. –  Baarn Sep 1 '12 at 15:47
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2 Answers

up vote 5 down vote accepted

The equation for total aerodynamic drag force on a bicycle is well-understood and, you're right, air density has a role in it: the higher the density, the greater the drag. As you guessed, density decreases with increasing temperature so warm air is less dense than cold air. It's also probably no surprise that density decreases with pressure. However, evidently counter-intuitively, humid air is less dense than dry air. The simplest explanation is that water molecules contain two hydrogen atoms and one oxygen atom so they have a molecular mass of 18 g/mol, while dry air has a molecular mass of around 29 g/mol. You can check the wikipedia page on air density for more details.

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Air density alone has a fairly substantial impact on speed, if all else is equal (and if my calculations are correct!)

Based on the equation and from this page on cycling aerodynamics, and the air-density values from Wolfram Alpha, I came up with:

At 300 watts, at  0*C, will travel at 39.34 km/h
At 300 watts, at 10*C, will travel at 40.06 km/h
At 300 watts, at 20*C, will travel at 40.77 km/h
At 300 watts, at 30*C, will travel at 41.46 km/h
At 300 watts, at 40*C, will travel at 42.14 km/h

That's based on the "in the drops" drag-coefficient/frontal-area values, at a consistent level above sea (I think at sea-level, not certain)

Put another way:

To travel at 13.8m/s in 0*C requires  478.45 watts
To travel at 13.8m/s in 10*C requires 461.43 watts
To travel at 13.8m/s in 20*C requires 445.52 watts
To travel at 13.8m/s in 30*C requires 430.72 watts
To travel at 13.8m/s in 40*C requires 417.03 watts

(13.8m/s is about 50km/h, arbitrary number)

To "show my working", here's the Python script I used to calculate the above:

#!/usr/bin/env python2
"""Impact on air-density on cycling speeds

Written in Python 2.7
"""


def Cd(desc):
    """Coefficient of drag

    Coefficient of drag is a dimensionless number that relates an
    objects drag force to its area and speed
    """

    values = {
        "tops": 1.15, # Source: "Bicycling Science" (Wilson, 2004)
        "hoods": 1.0, # Source: "Bicycling Science" (Wilson, 2004)
        "drops": 0.88, # Source: "The effect of crosswinds upon time trials" (Kyle,1991)
        "aerobars": 0.70, # Source: "The effect of crosswinds upon time trials" (Kyle,1991)
        }
    return values[desc]


def A(desc):
    """Frontal area is typically measured in metres squared. A
    typical cyclist presents a frontal area of 0.3 to 0.6 metres
    squared depending on position. Frontal areas of an average
    cyclist riding in different positions are as follows

    http://www.cyclingpowermodels.com/CyclingAerodynamics.aspx
    """

    values = {'tops': 0.632, 'hoods': 0.40, 'drops': 0.32}

    return values[desc]


def airdensity(temp):
    """Air density in kg/m3
    Values are at sea-level (I think..?)

    Values from changing temperature on:
    http://www.wolframalpha.com/input/?i=%28air+density+at+40%C2%B0C%29

    Could calculate this:
    http://en.wikipedia.org/wiki/Density_of_air
    """
    values = {
        0: 1.293,
        10: 1.247,
        20: 1.204,
        30: 1.164,
        40: 1.127,
        }

    return values[temp]


"""
F = CdA p [v^2/2]
where:
F = Aerodynamic drag force in Newtons.
p = Air density in kg/m3 (typically 1.225kg in the "standard atmosphere" at sea level) 
v = Velocity (metres/second). Let's say 10.28 which is 23mph
"""


def required_wattage():
    """What wattage will the mathematicallytheoretical cyclist need to
    output to travel at a specific speed?
    """

    position = "drops"

    for temp in (0, 10, 20, 30, 40):
        v = 13.8 # m/s
        F = Cd(position) * A(position) * airdensity(temp) * ((v**2)/2)
        watts = v*F
        print "To travel at %sm/s in %s*C requires %.02f watts" % (v, temp, watts)


def speed_from_wattage():
    """Given a specific force output, how fast will a
    mathematicallytheoretical cyclist travel?
    """
    from math import sqrt

    position = "drops"

    for temp in (0, 10, 20, 30, 40):
        # Calculate some reasonable number for F... I think..? Made
        # sense when I wrote it, but now slightly confusing
        v = 13.8 # m/s
        watts = 300
        F = watts/v # because watts=v*F

        # "F = CdA p [v^2/2]" solved for "v" with sympy:
        """
        from sympy import symbols, solve, Eq
        F, Cd, A, airdensity, v = symbols("F Cd A airdensity v")
        solve(Eq(F, Cd * A * airdensity * (v**2/2)), v)
        """

        v = sqrt(2) * sqrt(F / (Cd(position) * A(position) * airdensity(temp)))
        v_in_km_h = ((v * 60*60)/1000)

        print "At %s watts, at %d*C, will travel at %.02f km/h" % (
            watts, temp, v_in_km_h)


if __name__ == '__main__':
    speed_from_wattage()
    required_wattage()
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Nice, though you're missing the contribution to total power demanded by rolling resistance, which will be around Crr*mass*g*v where g ~= 9.8 m/sec^2. It appears the Wolfram air density calculation is assuming some barometric pressure and humidity since you didn't put one in. And, as an aside, I recommend method #4 on that cyclingpowermodels.com page. –  R. Chung Oct 16 '12 at 13:50
    
@R.Chung Hm, good point.. but shouldn't matter much in this case, right? I guess the rolling resistance would be a constant value added to the required-power, and the default barometric-pressure/humidity values are fine and arbitrary (without knowing the "Olympic mandated" values) –  dbr Oct 16 '12 at 15:31
1  
Oh, I was just commenting on your calculation of the power demand. At the speeds you were using in your example, rolling resistance will account for ~50 watts or so. OTOH, if you were just looking at differences in power demand with air density, it's easier to notice that an x% change in air density translates to exactly the same x% difference in aero drag. –  R. Chung Oct 16 '12 at 16:23
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